Light will not travel on a geodesic in the background spacetime, as light is an electromagnetic field and has stress-energy and so itself warps the spacetime around it. However if the light induced curvature is small compared to the curvature already there, then it won't change things much, so will go on an approximately geodesic route.
Same for you spaceship and all it's fuel. But again, as long as the ship induced curvature is not large compare to the curvature without the ship the changes are not large. And again, the comparison is ship-induced curvature compare to the curvature without the ship, it has nothing to do with the fueled up energy versus empty ship energy, well as long as the fueled up ship is the one that doesn't change the curvature much.
OK, let's get to that fuel required. Imagine two ships. The Space Valdez and the Space Chicken. One time in Camp Music, in the past, the Space Valdez is manufactured at Music Camp. It harvests fuel at Music Camp, way more fuel than it personally needs and plans to transport some fuel to Camp Parts Unknown, which is far away. So far away that the Space Valdez carries factories capable of manufacturing and assembling replacement spaceship parts. The occupants of Music Camp wish to be good galactic occupants so instruct the Space Valdez to respond with overwhelming kindness to distress broadcasts even if it uses up much fuel and resources. Who knows maybe there is a default way to program fuel transports to react so that others can recognize that it isn't a giant missile and maybe making it respond to any request to fuel or manufacture is a way to do that. Not saying that it is smart, but say it happened, even just once, in the history of the universe. So the Space Valdez sets off on its maiden voyage, fueled up, with manufacturing and communications capability and full of goodwill. Aims towards Camp Parts Unknown and goes.
So the Space Valdez is transporting a huge amount of fuel taking a nice geodesic through spacetime when it encounters a distress broadcast from a decrepit and low fuel ship called the Space Chicken. The Space Valdez accelerates to join and matches the accelerating trajectory of the Space Chicken and exchanges vast data back and forth at extremely close range. This encounter is considered productive and the Space Valdez uploads a datastore from the Space Chicken that describes an earlier state of the Space Chicken in a surprising similar looking region of spacetime. It repairs, refuels and wholesale rebuilds the Space Chicken as needed. Including resetting the whole datastore of the Space Chicken. Then they bid farewell and the Space Valdez heads back out to Camp Unknown.
Meanwhile the rejuvinated Space Chicken is in bootup mode and runs on autopilot mode which includes accelerating hard to test activate the rockets. It saves a checkpoint in the datastore (that includes instructions about how to build and repair the Space Chicken) as part of bootup process. It then uses the newly acquired fuel and polished craft to hurtle around the cylinder on an accelerated trajectory. And it made the log of it's current state shortly after losing communication distance with the Space Valdez. Eventually battered and low on fuel the Space Chicken, so very damaged but having always maintained a good record (at cost to degredation of the rest of the ship and using up of fuel) of it's earlier state and keeping the communication system good and full of data of the kind that strangers might find beneficial to have, the Space Chicken starts to broadcast a distress broadcast. This transmission is picked up by the recently manufactured Space Valdez on its first voyage.
OK, so we have the complete picture. All the fuel that is shot out of the Space Chicken's engines can be traced back as having come from the Space Valdez and then originally from Camp Music. Same with all the wear and tear energy that inevitable happened as the Space Valdez fixed up the Space Chicken. The information about how to manufacture, repair, build, program, and reboot the Space Chicken are a type of Djinn, with no ultimate source, but merely flow from one region of spacetime to another. In every local region, the physics is normal. Parts wear out, systems are used to maintain redundancy, material is used to protect the integrity of data, energy and momentum are shot out one way to accelerate the other way sop fuels goes down except when being refueled in flight by a fuel transport vessel. Nothing weird at all. Particularly no violations of conservation of energy, because that is a local law so we just need small coordinate patches and in each patch the energy changes by exactly how much flowed in. This scenario solves that fine. We could write a chapter book about each region of spacetime over an small bit of space and a small bit of time and the story looks perfectly ordinary.
As for entropy. In a sense the same thing happens, at each point entropy is already there and as time goes on the local entropy goes down by an amount less than the entropy that flows out. The Space Chicken uses fuel to do things and emits waste heat during the whole trip. With local stores of entropy, it is again a local law.
With no quantum, there is no problem. And no violation of laws of energy or thermodynamics. Other answers that mention things vanishing and appearing are wrong. General Relativity is a local theory, just break it up into small pieces where nothing weird happens, if you can't do that then you've done it wrong. If when you glue it all together you look at it and you don't like it, that's your problem if each local piece followed the rules just fine. And you don't ever even need to bring up words like forever. Just break it into pieces and describe each piece in a totally normal way and make it clear how the pieces line up.
I hope I did that for you in this example, and if you break it into pieces there is never a need for the word forever, and breaking it into pieces each of which looks fine, and sewing them together is what General Relativity is all about.
For your other questions:
Is it possible for a particle to be susceptible to external forces and still have zero rest mass?
A zero mass particle can still have its momentum changed by an external agent/interaction, and that can make it deviate from a geodesic, but unlike for a massive particle it need not, it could just increase its energy without changing direction so could continue the same direction.
Is it possible that Chandrasekhar and Wright were actually right in suggesting that Gödel's philosophical conclusions are questionable, and that they hit the nail on the head by focusing on the geodesics in the Gödel's universe?
The focus on geodesics is a red herring and a wild goose chase. Nothing travels on a geodesic unless it has no energy, no stress, no momentum and no spin. Plus you can refuel so there is no issue whatsoever. There are problems with Closed Timelike Curves, but they are more about quantum effects or stability issues.